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Equivalence of \(L_ p\) stability and exponential stability for a class of nonlinear semigroups. (English) Zbl 0547.47041

Let S(t) be a strongly continuous semigroup of bounded linear operators on a Banach space Y. A. Pazy has shown that if \(p\geq 1\) the p- integrability over [0,\(\infty)\) of S(t)y for any y in Y is equivalent to the exponential stability of S(t). R. Datko has then extended this to two-parameter semi-groups of bounded linear operators. In this paper a similar equivalence of \(L_ p\)-stability with \(p>0\) and exponential stability is shown for a certain class of nonlinear semigroups. It contains semilinear deterministic and stochastic evolution equations with Lipschitz nonlinearities. A possible application of this equivalence to obtain weaker sufficient conditions for exponential stability is discussed by a simple example.

MSC:

47H20 Semigroups of nonlinear operators
93D20 Asymptotic stability in control theory
93E15 Stochastic stability in control theory
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